scholarly journals Accurate discretization of poroelasticity without Darcy stability

2021 ◽  
Author(s):  
Kent-Andre Mardal ◽  
Marie E. Rognes ◽  
Travis B. Thompson
Keyword(s):  
Hydraulic Conductivity ◽  
Stability And Convergence ◽  
Convergence Properties ◽  
Basic Premise ◽  
Storage Coefficient ◽  
Galerkin Discretization ◽  
Discretization Schemes ◽  
Biot’S Equations ◽  
Discrete Spaces

AbstractIn this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.

scholarly journals Decentralized controllers for shape generation with robotic swarms

Robotica ◽  
2008 ◽  
Vol 26 (5) ◽  
pp. 691-701 ◽  
Author(s):  
M. Ani Hsieh ◽  
Vijay Kumar ◽  
Luiz Chaimowicz
Keyword(s):  
Relative Distance ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Geometric Pattern ◽  
Closed Curves ◽  
State Information ◽  
Robotic Swarms ◽  
Planar Curve ◽  
Decentralized Controllers

SUMMARYWe address the synthesis of controllers for a swarm of robots to generate a desired two-dimensional geometric pattern specified by a simple closed planar curve with local interactions for avoiding collisions or maintaining specified relative distance constraints. The controllers are decentralized in the sense that the robots do not need to exchange or know each other's state information. Instead, we assume that the robots have sensors allowing them to obtain information about relative positions of neighbors within a known range. We establish stability and convergence properties of the controllers for a certain class of simple closed curves. We illustrate our approach through simulations and consider extensions to more general planar curves.

scholarly journals FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS

2012 ◽  
Vol 17 (2) ◽  
pp. 227-244 ◽  
Author(s):  
Natalia Boal ◽  
Francisco Jos´e Gaspar ◽  
Francisco Lisbona ◽  
Petr Vabishchevich
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
System Of Equations ◽  
Convergence Properties ◽  
Discrete Energy ◽  
Numerical Resolution ◽  
Multigrid Convergence

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.

scholarly journals A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

2021 ◽  
pp. 58-69
Author(s):  
Lawrence Osa Adoghe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Stiff Systems ◽  
Convergence Properties ◽  
Main Method ◽  
The Stability ◽  
Third Derivative

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

scholarly journals Formation Control of Robotic Swarm Using Bounded Artificial Forces

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Long Qin ◽  
Yabing Zha ◽  
Quanjun Yin ◽  
Yong Peng
Keyword(s):  
Formation Control ◽  
Control Algorithm ◽  
Stability And Convergence ◽  
Multirobot Systems ◽  
Convergence Properties ◽  
Swarm Behavior ◽  
Detailed Simulation ◽  
Robotic Swarm ◽  
The Stability ◽  
Significant Attention

Formation control of multirobot systems has drawn significant attention in the recent years. This paper presents a potential field control algorithm, navigating a swarm of robots into a predefined 2D shape while avoiding intermember collisions. The algorithm applies in both stationary and moving targets formation. We define the bounded artificial forces in the form of exponential functions, so that the behavior of the swarm drove by the forces can be adjusted via selecting proper control parameters. The theoretical analysis of the swarm behavior proves the stability and convergence properties of the algorithm. We further make certain modifications upon the forces to improve the robustness of the swarm behavior in the presence of realistic implementation considerations. The considerations include obstacle avoidance, local minima, and deformation of the shape. Finally, detailed simulation results validate the efficiency of the proposed algorithm, and the direction of possible futrue work is discussed in the conclusions.

scholarly journals Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Eleonora Messina ◽  
Antonia Vecchio
Keyword(s):  
Integral Equations ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Stability And Convergence ◽  
Time Behavior ◽  
Convergence Properties ◽  
Convergence Of Solutions ◽  
Long Time ◽  
The Stability

We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

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